PPP Rules, Macroeconomic (In)stability and Learning

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Abstract:

Governments in emerging economies have
pursued real exchange rate targeting through Purchasing Power
Parity (PPP) rules that link the nominal depreciation rate to
either the deviation of the real exchange rate from its long run
level or to the difference between the domestic and the foreign
CPI-inflation rates. In this paper we disentangle the conditions
under which these rules may lead to endogenous fluctuations due to
self-fulfilling expectations in a small open economy that faces
nominal rigidities. We find that besides the specification of the
rule, structural parameters such as the share of traded goods (that
measures the degree of openness of the economy) and the degrees of
imperfect competition and price stickiness in the non-traded sector
play a crucial role in the determinacy of equilibrium. To evaluate
the relevance of the real (in)determinacy results we pursue a
learnability (E-stability) analysis for the aforementioned PPP
rules. We show that for rules that guarantee a unique equilibrium,
the fundamental solution that represents this equilibrium is
learnable in the E-stability sense. Similarly we show that for PPP
rules that open the possibility of sunspot equilibria, a common
factor representation that describes these equilibria is also
E-stable. In this sense sunspot equilibria and therefore aggregate
instability are more likely to occur due to PPP rules than
previously recognized.

*Email address: luis-felipe.zanna@frb.gov. This paper is based on
a chapter of my Ph.D. dissertation at the University of
Pennsylvania. I am grateful to Martín Uribe, Stephanie
Schmitt-Grohé and Frank Schorfheide for comments on early
versions of this paper. I also thank David Bowman, George Evans and
Dale Henderson for comments and suggestions. All errors remain
mine. The views in this paper are solely the responsibility of the
author and should not be interpreted as reflecting the views of the
Board of Governors of the Federal Reserve System or of any other
person associated with the Federal Reserve System. Return to text

1 Introduction

It has been claimed that the real exchange rate is perhaps the
most popular real target in developing economies. The reason is
that policy makers in these economies are always concerned about
avoiding losses in competitiveness in foreign markets, or
similarly, about maintaining purchasing power parity (PPP). In
order to achieve the real exchange target policy makers often
follow PPP rules. Such rules link the nominal rate of devaluation
of the domestic currency to the deviation of the real exchange rate
from its long run level or to the difference between the domestic
inflation rate and the foreign inflation rate. For instance, Calvo
et al. (1995) argued that Brazil, Chile and Colombia followed such
rules in the past.

The characterization of the channels through which real exchange
rate targeting affects the business cycles in emerging economies is
a central issue in the design and implementation of the PPP rules.
The theoretical literature about PPP rules has tried to disentangle
these channels.1 One of
these important attempts is made by Uribe (2003) who analyzes a PPP
rule whereby the government increases the devaluation rate when the
real exchange rate is below its steady-state level. He pursues a
determinacy of equilibrium analysis and argues that PPP rules may
lead to aggregate instability in the economy by inducing endogenous
fluctuations due to self-fulfilling expectations.

From the economic policy-design perspective, this result has
important implications. It states that the aforementioned rules may
open the possibility of sunspot equilibria and lead the economy to
equilibria with undesirable properties such as a large degree of
volatility. This implication in turn suggests that a determinacy of
equilibrium analysis can be used to differentiate among rules
favoring those that at least avoid sunspot equilibria by
guaranteeing a unique equilibrium with a lower degree of
volatility.2 Although
appealing this argument is still far from complete and may suffer
from some drawbacks. The reason is that in the typical determinacy
of equilibrium analysis, it is implicitly assumed that agents can
coordinate their actions and learn the equilibria (unique or
multiple) induced by the rule. But relaxing this assumption may
have interesting consequences for the design of PPP rules. On one
hand, if agents cannot learn the unique equilibrium targeted by the
rule then the economy may end up diverging from this equilibrium.
But if this is the case then it is clear that there are some rules
that although guaranteeing a unique equilibrium, do not insure that
the economy will reach it.3 On the
other hand, if agents cannot learn sunspot equilibria then one may
doubt about the relevance of characterizing rules that lead to
multiple equilibria as ``bad'' ones. After all, if agents cannot
learn sunspot equilibria then they are less likely to occur.

Therefore, it seems clear that a determinacy of equilibrium
analysis of PPP rules should in principle be accompanied by a
learnability of equilibrium analysis. Both analyses would help
policy makers to distinguish and design PPP rules satisfying two
requirements: uniqueness and learnability of the equilibrium. The
first requirement would prevent the economy from achieving sunspot
equilibria with undesirable properties such as a large degree of
volatility. Whereas, the second requirement would guarantee that
agents can indeed coordinate their actions on the equilibrium the
policy makers are targeting.

The present paper is motivated by the interest of studying if
particular representations of the equilibria (unique or multiple)
induced by PPP rules are learnable in the Expectational - Stability
(E-Stability) sense proposed by Evans and Honkapohja (1999,
2001).45 In fact
our purpose in the present paper is three-fold. First we study and
disentangle the structural conditions of an open economy under
which an Uribe-type PPP rule may generate multiple equilibria (real
indeterminacy).6 We use a
small open economy model with traded and non-traded goods. We
assume flexible prices for the former and sticky prices for the
latter. Under this set-up we show how the aforementioned conditions
depend not only on the responsiveness of the rule to the real
exchange rate but also on some important structural parameters of
the economy. For instance we find that ceteris paribus, given the
sensitivity of the rule to the real exchange rate, the lower the
degree of openness of the economy (the lower the share of traded
goods), the more likely that the rule will induce aggregate
instability in the economy by generating multiple equilibria. In
addition, keeping the rest constant, the lower (the higher) the
degree of price stickiness (the degree of monopolistic competition)
in the non-traded sector, the more likely that the rule will lead
to real indeterminacy.

The second goal of this paper consists of showing that under
real determinacy the fundamental solution that describes the unique
equilibrium induced by the PPP rule is learnable in the E-stability
sense.7 In
addition we use the recent work by Evans and McGough (2003) to
prove that under real indeterminacy some common factor
representations of stationary sunspot equilibria are also
E-stable.8 This
result suggests that under some reasonable assumptions agents can
learn and coordinate their actions to achieve sunspot equilibria,
making them ``more likely'' to occur under PPP rules. In this sense
these equilibria should not be perceived as mere mathematical and
theoretical curiosities.

The natural question that arises from these results is whether
under a different timing of the PPP rule, it is possible for policy
makers to design a simple rule that avoids sunspot equilibria but
still induces a unique equilibrium whose characterization is
learnable. In accord with the findings in the interest rate rule
literature, we find that a PPP rule that is backward-looking in the
sense of being defined in terms of the (deviation of the) past real
exchange rate (from its long run level) satisfies these two
requirements.

Finally the third goal of this paper is associated with the
original work by Dornbusch (1980, 1982) that studies how a PPP rule
whereby the nominal exchange rate is linked to the (deviation of
the) current domestic price level (from its long-run level), may
affect the output price-level stability trade-off by playing a role
as an absorber of fundamental shocks.9 We analyze a rule motivated by
Dorbunsch's works assuming that the nominal devaluation rate is
positively linked to the difference between the domestic and
foreign CPI-inflation rates. In fact this specification tries to
capture the previously mentioned stylized facts about PPP rules in
Brazil, Colombia and Chile. As before we state the conditions under
which this rule leads to real indeterminacy. We also show that the
common factor representation of stationary sunspot equilibria as
well as the fundamental solution that describes the unique
equilibrium induced by the rule are learnable in the E-stability
sense.

The remainder of this paper is organized as follows. Section 2
presents the set-up of a sticky-price model with its main
assumptions. Section 3 pursues the determinacy of equilibrium
analysis for a PPP rule defined in terms of the current real
exchange rate. Section 4 deals with the learnability analysis for
the aforementioned rule. Section 5 pursues all the previous
analyses for a PPP rule defined in terms of the CPI-inflation rate.
Finally Section 6 concludes.

2 A Sticky-Price Model

2.1 The Household-Firm Unit

Consider a small open economy inhabited by a large number of
identical household-firm units indexed by
. The
household-firms live infinitely and the preferences of the
representative agent can be
described by the intertemporal utility function10

(1)

(2)

where
, and
denote the
consumption of traded and non-traded goods respectively,
and
are the
labor allocated to the production of the traded good and the
non-traded good.
denotes the expectational operator.11 Equations (1) and
(2) imply that the representative agent
derives utility from consuming traded and non-traded goods, and
from not working in either sector.

We assume that the non-traded good is a composite good. We
introduce monopolistic competition in the model by assuming that
the household-firm unit
can choose the price of the non-traded good it supplies,
, subject
to a particular demand constraint described by

(3)

where
represents the aggregate demand for the non-traded good and

We also assume that there are sticky prices in the production of
the non-traded good. This assumption is useful to understand the
last term of the intertemporal utility function (1). Following Rotemberg (1982) we suppose that the
household-unit dislikes having its price of non-traded goods grow
at a rate different from
, the
steady-state level of the non-traded goods inflation rate.12 We introduce sluggish adjustment in
prices not only because this will enrich our analysis, but also
because, as Uribe (2003) points out, one of the main reasons that
explains and motivates the real exchange targeting through PPP
rules, is that policy makers believe in eliminating the real
rigidities imposed by a fixed exchange rate system in an
environment with nominal rigidities.

The production of traded and non-traded goods only requires
labor and uses the following technologies

(4)

where
and are random productivity
parameters that satisfy

(5)

where
and
for
For simplicity
in the analysis we assume no correlation between the productivity
shocks.

The law of one price holds for the traded good and to simplify
the analysis we normalize the foreign price of the traded good to
one. Therefore, the domestic currency price of traded goods
() is equal to
the nominal exchange rate (
). This
simplification in tandem with (1) and
(2) can be used to derive the consumer price
index (CPI)

it is straightforward to derive the CPI-inflation rate,
, as a weighted
average of the nominal depreciation rate,
and the
inflation of the non-traded goods,
; that is

(8)

We define the real exchange rate () as the ratio between the price of traded goods
and the aggregate price of non-traded goods

(9)

From this definition of the real exchange rate we deduce
that

(10)

We assume that in each period the representative agent can purchase two types
of financial assets: fiat money
and
nominal state contingent claims,
that pay
one unit of currency in a specified state of period and that are traded
internationally. There exists a set of these state contingent
claims that completely spans the fundamental (intrinsic)
uncertainty associated with productivity shocks. Moreover following
Kimbrough (1986), we suppose that money reduces the transaction
costs in goods markets. These costs measured in terms of the traded
good can be described by

(11)

where
.

Using the previous assumptions the representative agent's flow
constraint each period can be written as13

(12)

where refers to the
period- price of a
claim to one unit of currency delivered in a particular state of
period divided by
the probability of occurrence of that state and conditional of
information available in period . Hence
denotes the cost of all
contingent claims bought at the beginning of period . Constraint (12)
says that the total end-of-period nominal value of the financial
assets can be worth no more than the value of the financial wealth
brought into the period, , plus non-financial income during the period net
of the value of taxes,
, the value of
consumption spending and the value of the liquidity transaction
costs.

To derive the period-by-period budget constraint of the
representative agent, it is important to notice that total
beginning-of-period wealth in the following period is given by

(13)

and that
corresponds to the price at period of a claim that pays one unit of currency in every
state in period and
represents the inverse of the risk-free gross nominal interest
rate, ; that is

(14)

Then we can use equations (12), (13) and (14) to derive the
budget constraint of the representative agent as

at all dates and under all contingencies, where represents the period-zero price
of one unit of currency to be delivered in a particular state of
period divided by
the probability of occurrence of that state, given information
available at time It is
given by

(17)

with
.

Under this sticky-price set-up the problem of the representative
agent is reduced to choose the sequences {
in order to maximize (1) subject to
(2), (3), (4), (11), (15) and (16), and given
and
and the
time paths for ,
,
, and and
Note that
the utility function specified in (1) and
(2) implies that the preferences of the agent
display non-sasiation. This means that constraints (15) and (16) both hold with
equality.

The first order conditions correspond to (15) and (16) both with
equality and

where
corresponds to the
multiplier of the budget constraint,
is the
multiplier associated with the demand constraint (3) and
.
The interpretation of the first order conditions is
straightforward. In particular, equation (18)
is the usual intertemporal envelope condition that makes the
marginal utility of consumption of traded goods equal to the
marginal utility of wealth (
)
multiplied by the intertemporal price of consuming traded
goods.14Condition
(19) implies that the marginal rate of
substitution between traded and non-traded goods must be equal to
the real exchange rate. In addition, condition (20) equalizes the marginal revenue products of labor
among sectors. Equation (21) represents the
demand for real balances of money as an increasing function of
consumption expenditure and a decreasing function of the risk-free
nominal interest rate. And finally condition (22) implies a standard pricing equation for
one-step-ahead nominal contingent claims. Note that in each period
there is one
condition of this type for each possible state in period

Finally we postpone the explanation of condition (23). The reason is that it will be used to derive the
augmented Phillips curve for non-traded goods, that is actually one
of the relevant equations for the determinacy and learnability of
equilibrium analyses.

2.2 The Government

The government issues two nominal liabilities: money,
, and state
contingent debt
It also
levies taxes, pays
interest on its debt, and receives revenues from
seigniorage Thus we can
write the government budget constraint as

(24)

where
The government
follows a Ricardian fiscal policy. That is, it picks the path of
taxes,
satisfying the intertemporal version of (24)
in conjunction with the transversality condition

(25)

Finally we define the monetary policy as in Uribe (2003). The
government follows a PPP rule whereby the government sets the
nominal devaluation rate as a function of the deviation of the
current real exchange rate () with respect to its long-run level
(). That is

(26)

where is a continuous
function that in steady state satisfies
.

2.3 The Equilibrium

We will focus on a symmetric
equilibrium in which all the household-firm units choose the
same price for the good they produce. Therefore in equilibrium all
agents are identical and we can drop the index . In equilibrium the money market and
the non-traded goods market clear. Thus

(27)

and
As usual we ignore the wealth effects due to inflation by assuming
that the transaction liquidity costs, , are rebated to the
representative agent in a lump-sum fashion.

We also assume free capital mobility. This implies that the
following non-arbitrage condition must hold

(28)

where
refers to the period-
foreign currency price of a claim to one unit of foreign currency
delivered in a particular state of period divided by the probability of
occurrence of that state and conditional of information available
in period . An
equivalent condition to (22) holds for the
foreign economy (rest of the world). That is,

(29)

where
and
represent
the marginal utility of wealth, the price of traded goods and the
subjective discount rate in the foreign economy respectively. Using
(22), (28), (29), the law of one price for traded goods and the
assumption that
we
can derive that
that holds at all dates and under all
contingencies.15 This
equation implies that the domestic marginal utility of wealth is
proportional to its foreign counterpart. Then
where refers to a constant parameter
that determines the wealth difference across countries. Since we
are dealing with a small open economy,
can be taken as an exogenous variable. To simplify the analysis we
assume that
is
constant and equal to
. This
assumption implies that
becomes a
constant. Consequently

(30)

But this result of a constant marginal utility and conditions
(14 ) and (22) imply
that

(31)

where denotes the
expectation operator. Note that condition (31)
is very similar to the uncovered interest parity condition.

Utilizing (20), (23),
the symmetry in
equilibrium, the equilibrium condition in the non-traded sector (
)
and (30), we obtain

(32)

that corresponds to the augmented Phillips curve for the non-traded
goods inflation.16

Furthermore applying the symmetry in equilibrium and recalling
(30), we can rewrite (18), (19) and (21) as

(33)

(34)

(35)

We proceed giving the definition of a symmetric equilibrium for
a government that pursues a Ricardian fiscal policy and follows a
PPP rule that responds to the current real exchange rate as
described by (26).

Definition 1Given,andand the exogenous
stochastic processesa Symmetric Equilibrium under a Ricardian
fiscal policy is defined as a set of stochastic processessatisfying conditions (32),
(33), (34), (35), the intertemporal version of (24) together with (25), the PPP
rule defined by (26), the money market
clearing condition (27), definitions (7), (17) and equations (10), (14), and (31).

3 The Determinacy of Equilibrium
Analysis

To pursue the determinacy of equilibrium analysis we reduce the
model further. To do so we can use conditions (33) and (34) to obtain

(36)

that together with the PPP rule (26) and
equations (5), (10), (31) and (32), are the only equations necessary to pursue the
determinacy of equilibrium analysis in our model. They help us to
find the stochastic processes
These set of
equations are also useful to define the non-stochastic steady
state. It corresponds to

where
and
denotes
the long-run nominal devaluation rate that is determined by the
government.

We point out that we do not need to consider in the determinacy
analysis equations (24) and (25). The reason is that under a Ricardian fiscal
policy, the intertemporal version of the government's budget
constraint in conjunction with its transversality condition will be
always satisfied. Moreover the stochastic processes
can be recovered
using (7), (14),
(17), (27), (34) and (35).17

We can go further reducing and log-linearizing the model. Using
equations (5), (10), (26), (31), (32), and (36) yields18

(37)

(38)

(39)

(40)

where
, represents the steady-state
level of ,

(41)

and
For our future analyses it is important to observe that
and
To see this
recall our assumptions about the values that are feasible to assign
to the structural parameters of the model and the definition of the
steady-state.

As we mentioned before in this analysis we only study the
possibilities of real indeterminacy or
real determinacy of the equilibrium. By
real indeterminacy we mean a situation
in which the behavior of one or more (real) variables of the model
are not pinned down by the model. This situation implies that there
are multiple equilibria and opens the possibility of the existence
of sunspot equilibria.

Before we analyze the conditions under which PPP rules may lead
to real indeterminacy, it is worth constructing some intuition
using the model of why these rules may induce equilibria in which
expectations are self-fulfilled. In order to accomplish this task
we can assume perfect foresight (no uncertainty). Then we rewrite
equations (37) and (38) as

Equation (43) implies that current
inflation of non-traded goods is determined by the discounted sum
of the expected future real exchange rates and nominal depreciation
rates. The first term inside of the parenthesis is associated with
future real exchange rates. It captures the fact that higher
expected future real exchange rates make non-traded goods become
relatively cheaper than traded goods. This leads to a higher
expected future excesses of demand for non-traded goods to which
the firm-unit responds raising the current price of non-traded
goods up and therefore increasing the current non-traded goods
inflation rate. On the other hand, the second term in (43) that is associated with future nominal
depreciation rates captures the effect of the intertemporal price
of consumption on the determination of the current non-traded goods
inflation. In essence, expectations of nominal appreciation
(negative nominal depreciation rates) decrease the nominal interest
rate provided that the uncovered interest parity condition holds
under perfect foresight. But a decrease in the nominal interest
rate pushes the liquidity transaction costs down, which in turn
expands consumption of non-traded (and traded) goods. This increase
in consumption lead to a positive excess of demand for non-traded
goods and therefore to a higher current inflation.

Equation (44) simply describes the
depreciation (or appreciation) of the real exchange rate as a
difference between the nominal depreciation rate and the non-traded
goods inflation rate.

With these two last equations, equation (42) and the PPP rule,
we can show that multiple equilibria are possible by
constructing a self-fulfilling equilibrium. Assume that at time
when the economy
is in its steady state, private agents expect a real appreciation
after time . More
explicitly they expect the following path for the (deviation of
the) real exchange rate.20 At time
the real exchange
rate is at the steady state level (
). At
times and the real exchange rate is above
its steady state level (
and
)
and satisfies
showing a real
appreciation and convergence to the steady state level over
time Given this path of
the real exchange rate and given the PPP rule, the government will
induce a nominal appreciation (
with
and
) over time
after time . Then using
our interpretation of (43) we can infer that
the expected path for the real exchange rate and the expected
nominal appreciation will motivate the household-firm unit to raise
the price of non traded goods in period . In other words inflation at time
will go up (
). By equation (42) this effect will increase inflation of non
traded goods in period
(
), if it is strong enough to
overcome the opposite effects that the assumed path for the real
exchange rate (
) and
the rule-induced path for the nominal depreciation rate (
) have over inflation of
non-traded goods at period (
).
Observe that this possibility is determined by the values of the
structural parameters of the model that affect and , and by the value of the nominal depreciation
response coefficient to the real exchange rate (). But if inflation of
non-traded goods goes up (
) and people expect a nominal
appreciation (
) in period accordingly with equation
(44), we conclude that the real exchange
rate will appreciate over time ((
). Since all the
variables of the system, including the real exchange rate, converge
to their steady state level over time then the original
expectations of a future real appreciation will be
self-fulfilled.

Although this intuitive argument points out the possibility of
self-fulfilling equilibria induced by a PPP rule, it is important
to disentangle the conditions under which these equilibria are
possible. The following proposition achieves this goal
characterizing locally the equilibrium for the model described by
equations (37)-(40).

Proposition 1Suppose the government follows a PPP rule that is
described by
with
Let and be defined as in (41) and
define

a)Ifandthen there is real indeterminacy.

b)Ifandthen there is real determinacy.

c)Ifthen there is real determinacy for any.

Proof. See Appendix.

From Proposition 1 it is clear that conditions under which PPP
rules lead to multiple equilibria do not simply depend on the
response coefficient
On the
contrary some of the structural parameters of the model play a
fundamental role in the determinacy of equilibrium. In essence all
the parameters that affect and are
relevant for the analysis. In order for the PPP rule to induce
multiple equilibria two conditions must be satisfied. The first one
constrains the possible values that the structural parameters may
take (
); the second one points out the
importance of the PPP rule on inducing real indeterminacy. It sets
a threshold for the nominal depreciation response coefficient that
depends on the structural parameters of the model (
). Even more interesting
is the result of part c) in the
proposition. It says that for some values that the structural
parameters may take and regardless of the value of the nominal
devaluation response coefficient (, the model displays real determinacy. This
result contrasts with the results of Uribe (2003) that claims that
if the elasticity of the PPP rule is sufficiently large then a
model with sticky-prices always displays real indeterminacy.

To understand the important role that some of the structural
parameters of the model may play in the determinacy of equilibrium
analysis, we study how the aforementioned threshold (
)
varies with respect to some of these structural parameters.
Specifically we consider the share of traded goods (), the degree of monopolistic
competition in the non-traded sector () and the degree of price
stickiness in the non-traded sector ().21 Note
that the share of traded goods can be considered a measure of the
degree of openness of the economy with
describing a very closed economy. The following corollary
summarizes the main results.

Corollary 1Suppose the government follows a PPP rule given bywith. Letbe defined as in Proposition 1 and
assume,
thena); b)andc).

Proof. See Appendix.

Using Proposition 1 and Corollary 1 we can understand the
effects of varying some of the structural parameters and the
semi-elasticity of the rule (
on
the determinacy of equilibrium. In fact when
we can conclude that given the semi-elasticity of the PPP rule, the
less open the economy is (the lower is), the more likely that the PPP rule will
induce aggregate instability in the economy by generating multiple
equilibria. In addition, given the semi-elasticity of the PPP rule
and keeping the rest constant, the higher the degree of
monopolistic competition in the non-traded sector (the higher
), the more likely
that the rule will lead to real indeterminacy. Finally under
ceteris paribus and given the semi-elasticity of the rule we find
that the lower the degree of price stickiness in the non-traded
sector (the lower ),
the more feasible that the rule will induce multiple
equilibria.

Notwithstanding the relevance of these analytical results, it is
crucial to investigate their quantitative importance. To accomplish
this we rely on a specific parametrization of the model. Since this
exercise is merely indicative we borrow some values of the
parameters from previous studies about emerging and small open
economies.22
Following Schmitt-Grohé and Uribe's (2001) study about
Mexico we assign the following values to some of the relevant
structural parameters of the model:
per
quarter,
per quarter,
,
and
. We
set
that
corresponds roughly to the imports to GDP share in Mexico during
the 90's. Finally we set , that corresponds to Dib's (2001) estimate
of for Canada in a
model with only nominal rigidities.23 With these values, we will perform
four exercises characterizing locally the equilibrium. We will vary
the semi-elasticity of the rule,
and
one and only one of the following structural and policy parameters:
, and
We
summarize the parametrization in the following table.

Table 1

The results of our exercises are presented in Figure 1, where
``I'' stands for real indeterminacy and ``D'' stands for real
determinacy. As can be observed, this figure confirms the results
in Proposition 1 and Corollary 1 showing how significant these
results are in quantitative terms. Consider the top left panel.
From this panel we can infer the following. Suppose that the
government in response to a 1 per cent appreciation of the real
exchange rate, devalues the nominal exchange rate by 2 percent. In
other words, assume that the semi-elasticity of the PPP rule is -2.
Whereas this PPP rule may induce multiple equilibria in an economy
whose degree of openness is 0.2, the same rule leads to a unique
equilibrium in an economy whose degree of openness is 0.6.

Similar inferences can be pursued from the top right and bottom
left panels of Figure 1. That is although a rule with
semi-elasticity of -2 guarantees a unique equilibrium in an economy
with a degree of monopolistic competition of 5 (a degree of price
stickiness of 5), the same rule induces multiple equilibria when
the aforementioned degree corresponds to 15 (2).

Although it is not possible to derive an analytical result to
see how varying the implied nominal depreciation target (
) and
the semi-elasticity of the rule (
) affects the determinacy of
equilibrium, it is possible to evaluate this quantitatively as
presented in the bottom right panel of Figure 1. This panel
illustrates that given the semi-elasticity of the rule the lower
the implied nominal depreciation target the more likely is that the
PPP rule will induce real indeterminacy. In addition, it is
important to observe that in all four panels of Figure 1 there are
regions for which the model always displays real determinacy
regardless of the semi-elasticity of the rule. This agrees with
part c) of Proposition 1.

To finalize this section we want to point out that similar
qualitative results to the ones presented in this section can be
obtained if the PPP rule is defined in terms of the real
depreciation rate. That is
where
24

4 The Learnability Analysis

The importance of the result from the previous section, that a
PPP rule may induce aggregate instability by generating multiple
equilibria in the economy, stems from the fact that such rule opens
the possibility of expectations driven fluctuations in economic
activity. In particular, the model may admit self-fulfilling
rational expectations equilibria driven by extraneous processes
known as sunspots.25

However the previous results, as the ones in Uribe (2003), do
not discuss the attainability of these PPP rule induced sunspot
equilibria. They do not even mention how attainable the unique
equilibrium is. Strictly speaking, and regardless of real
determinacy or real indeterminacy, it is not clear whether and how
agents may coordinate their actions in order to achieve a
particular equilibrium in the model. The purpose of this section is
to address this issue. We want to study the potential of agents to
learn the unique equilibrium characterized by the fundamental
solution and sunspot equilibria described by a common factor
solution.

Figure 1: This figure shows
how the local determinacy of equilibrium varies with respect to the
semi-elasticity of the rule (
), the
share of traded goods (), the degree of monopolistic competition in
the non-traded sector (), the degree of price stickiness in the
non-traded sector ()
and the implied nominal depreciation rate target (
). ``I''
stands for real indeterminacy (multiple equilibria) and ``D''
stands for real determinacy (a unique equilibrium). ``ES''
corresponds to E-Stability.

Description of Figure 1

Figure 1 shows the combinations of the
semi-elasticity of the rule and other structural parameters of the
model under which there is a unique equilibrium (real determinacy)
whose Minimal State Variable (MSV) representation is learnable in
the E-stability sense. These combinations are denoted by `` D-ES''.
The figure also shows the combinations of these parameters under
which there are multiple equilibria (real indeterminacy) and
sunspot equilibria whose Common Factor (CF) representation is
learnable in the E-stability sense. We denote these combinations by
`` I-ES''. Figure 1 has four panels. We proceed to describe each
panel. The top-left panel shows the combinations for the
semi-elasticity of the rule (
) and the
share of traded goods () under which there is `` D-ES'' or `` I-ES''.
The panel measures
on the
vertical axis with
and on the horizontal axis with
For
and
regardless of
there is
always `` D-ES''. For
there is a
frontier of combinations of
and
that separates
the combinations of `` D-ES'' from `` I-ES''. This frontier is
concave and has a negative slope. It crosses the
-axis at
-1.5 and the -axis at
0.7 approximately.(`` approximately'' since cannot be zero.) To the
north-east of this frontier we have the combinations for
and
under which there
is `` D-ES''. To the south-west of this frontier we have the
combinations for
and
under which there
is `` I-ES''.

The top-right panel shows the combinations for the
semi-elasticity of the rule (
) and the
degree of monopolistic competition () under which there is `` D-ES'' or `` I-ES''. This
panel measures
on the
vertical axis with
and on the horizontal axis with
For
and
regardless of
there is
always `` D-ES''. For there is a frontier of combinations of
and
that separates the
combinations of `` D-ES'' from `` I-ES''. This frontier is concave
and has a positive slope. It crosses the -axis.at 6 approximately. To the
north-west of this frontier we have the combinations for
and
under which there is
`` D-ES''. To the south-east of this frontier we have the
combinations for
and
under which there is
`` I-ES''.

The bottom-left panel shows the combinations for the
semi-elasticity of the rule (
) and the
degree of price stickiness () under which there is `` D-ES'' or `` I-ES''.
This panel measures
on the
vertical axis with
and on the horizontal axis with
For
and
regardless of
there is
always `` D-ES''. For there is a frontier of combinations of
and
that separates the
combinations of `` D-ES'' from `` I-ES''. This frontier is concave
and has a negative slope. It crosses the
-axis at
-0.8 and the -axis.at
4.75 approximately.(`` approximately'' since cannot be zero.) To the
north-east of this frontier we have the combinations for
and
under which there
is `` D-ES''. To the south-west of this frontier we have the
combinations for
and
under which there
is `` I-ES''.

The bottom-right panel shows the combinations for the
semi-elasticity of the rule (
) and the
depreciation target (
) under
which there is `` D-ES'' or `` I-ES''. This panel measures
on the
vertical axis with
and
on the
horizontal axis with
For
and regardless of
there is
always `` D-ES''. For
there is a frontier of combinations of
and
that
separates the combinations of `` D-ES'' from `` I-ES''. This
frontier is concave and has a negative slope. It crosses the
-axis at
0 and the
-axis at
0.022 approximately. To the north-east of this frontier we have the
combinations for
and
under
which there is `` D-ES''. To the south-west of this frontier we
have the combinations for
and
under
which there is `` I-ES''.

As a criterion of ``learnability'' of an equilibrium we will use
the concept of ``E-stability'' proposed by Evans and Honkapohja
(1999, 2001). That is, an equilibrium is ``learnable'' if it is
``E-Stable''.26
Consequently we start by assuming that agents in our model no
longer are endowed with rational expectations. Instead they have
adaptive rules whereby agents form expectations using recursive
least squares updating and data from the system. Then we derive the
conditions for expectational stability (E-stability).

In our analysis we will focus on the expectational stability
concept for the following reasons. First, in models that display a
unique equilibrium (real determinacy models), Marcet and Sargent
(1989) and Evans and Honkapohja (1999, 2001) have shown that under
some general conditions, the notional time concept of expectational
stability of a rational expectation equilibrium governs the local
convergence of real time adaptive learning algorithms. Specifically
they have shown that under E-stability, recursive least-squares
learning is locally convergent to the rational expectations
equilibrium. Second, Evans and McGough (2003) have numerically
argued that under some assumptions about the parameters of a linear
stochastic univariate model, with a predetermined variable, the
same argument applies when this model displays sunspot equilibria.
Formally they have stated that under a strict subset of the
structural parameter space, there exist stationary sunspot
equilibria that are locally stable under least square learning
provided that agents use a common factor representation for their
estimated law of motion.

We adapt the methodology of Evans and Honkapohja (1999, 2001)
and Evans and McGough (2003) to pursue the learnability
(E-stability) analysis. Accordingly we need to define the concept
of E-stability. In order to define it we give an idea of the
methodology we apply for the case of real determinacy.

To grasp the methodology, it becomes useful to reduce our model
to the following linear stochastic difference equations system. Use
(37), (38), (39) and (40) to rewrite the
model as

(45)

where

(46)

and denotes
in general (non-rational) expectations. Next, assume that the
agents follow a perceived law of motion (PLM) that in this case of
real determinacy corresponds to the fundamental solution27

Iterating forward this law of motion and using it to eliminate
all the forecasts in the model we can derive the implied actual law
of motion (ALM)

Then we obtain the T-mapping
whose fixed points correspond to the rational expectations
equilibria. An equilibrium is said to be E-stable if this mapping
is stable at the equilibrium in question. More formally a fixed
point of the T-mapping is E-stable provided that the differential
equation

is locally asymptotically stable at that particular fixed point,
where is defined as
the ``notional'' time.28

For the case of sunspot equilibria we apply the same methodology
but in that case the PLM is augmented by the sunspot and its
particular structure. In particular we will focus on the common
factor representation proposed by Evans and McGough (2003). Due to
space constraint we refer the readers to the aforementioned
references for a detailed explanation.29

It is important to observe that a fundamental part in the
learnability analysis consists of making explicit what agents know
when they form their forecasts. In the E-stability analysis
literature it is common to assume that when agents form their
expectations
they
do not know
In this
paper this assumption may be inconsistent with the assumptions that
we use to derive the equations of the model. In particular notice
that for the derivation of the first order conditions of the
representative agent we assume that
(or in a symmetric
equilibrium
) and
Therefore assuming in the learnability analysis that the agents do
not know when
forming expectations would have some implications for the
specification of the model. Specifically it would require to
replace
and
in
equations (37), (38)
and (40) with the expectations of
and
given
current information (
and
exogenous shocks). Henceforth for the learnability analysis of the
model (45) we will assume that when
forming expectations agents know

We proceed to present the results of the learnability analysis
for the fundamental solution of the model (45) in the following Proposition.

Proposition 2Suppose the government follows a PPP rule that is
described by
and
Let , and
be
defined as in (41) and (46)
respectively, and
.
Consider the following AR(1) representation

(47)

where
is defined as
a stable root of the quadratic equation and
is defined by
Under the real determinacy conditions
specified in Proposition 1, there is a unique equilibrium of the
model (45) characterized by the
fundamental solution (47), withand this solution is learnable in the
E-stability sense.

Proof. See Appendix.

Proposition 2 points out that when the model displays a unique
equilibrium (real determinacy) then the fundamental solution is
E-stable. This is the reason of denoting as ``D-ES'' the regions of
the four panels of Figure 1 for which the model displays not only
real determinacy but also E-stability. The importance of this
result stems from the fact that policy makers will face less
difficulties in implementing PPP rules that lead to a unique
equilibrium since they know that agents will coordinate on that
equilibrium and the macroeconomic system will not diverge away from
the targeted equilibrium.

It is also possible to show that under real indeterminacy the
fundamental solution or MSV solution can be E-stable. However in
this case policy makers will face other difficulties. In particular
under multiple equilibria there might be self-fulfilling rational
expectations equilibria driven by extraneous processes known as
sunspot. These equilibria may be characterized by undesirable
features such as larger volatility of macroeconomic variables
suggesting that policy makers should avoid rules that in principle
may induce multiple equilibria.

Although the previous argument may sound appealing, it may
suffer from some drawbacks. For instance, it is not clear whether
agents are able to coordinate their actions on a particular sunspot
equilibria. To clarify this issue the next proposition illustrates
that some particular representations of stationary sunspot
equilibria can be E-stable. To simplify the analysis and to be able
to derive analytical results we assume that

Proposition 3Suppose the government follows a PPP rule that is
described by
and
Let , and
be
defined as in (41) and (46)
respectively, and
, and
assume that
Consider the following common factor
representation

where,
are unique and correspond to the real roots of the
quadratic equationis arbitrary,andis a martingale difference sequence.30

a)Under the
real indeterminacy conditions specified in Proposition 1, there are
stationary sunspot equilibria of (45)
characterized by the common factor representation (48) and (49), where and
and this representation is learnable in the E-stability
sense.

b)Under the
real indeterminacy conditions specified in Proposition 1, there are
stationary sunspot equilibria of (45)
characterized by the common factor representation (48) and (49), where and
and this representation is NOT learnable in the
E-stability sense.31

Proof. See Appendix.

Proposition 3 demonstrates that some common factor
representations of sunspot equilibria induced by PPP rules are
learnable in the sense of E-stability. We would like to emphasize
the important role that the common factor representations proposed
by Evans and McGough (2003) play in the learnability analysis. To
see this, observe that the typical stationary sunspot equilibrium
representation,
where
denotes
the sunspot, is never E-stable. The reason is that such perceived
law of motion leads to an actual law of motion
that implies that
But this
suggests that the typical sunspot representation is not
learnable This
argument in tandem with Proposition 3 reveal that common factor
representations make stationary sunspot equilibria more likely to
arise under private learning than previously recognized.

Our results from the real determinacy and learnability of
equilibrium analyses pose the question of whether changing the
timing of the PPP rule avoids sunspot equilibria and still induces
a unique equilibrium that is E-stable. Similarly to the findings in
the interest rate rule literature, we find that a PPP rule that is
backward-looking in the sense of being defined in terms of the past
real exchange rate satisfies these two requirements.32 A backward-looking PPP rule can be
described as
and using this
specification and equations (37), (38) and (39), we can reduce
the model to

(50)

where

(51)

and denotes in
general (non-rational) expectations.

The following proposition summarizes the aforementioned
result.

Proposition 4Suppose the government follows a backward-looking PPP
rule that is described by
and
, and
consider the model described in (50) Let
, and
be
defined as in (41), and (51)
respectively and
.
Consider the AR(1) representation

(52)

where
is uniquely
defined as a stable root of the quadratic
equation
and is also
uniquely defined by

a)If
eitherandorandwiththen the model (50) displays a unique equilibrium (real
determinacy) that can be represented by the fundamental solution
(52) with. Moreover this
solution is E-stable.

b)Ifandthen there exists no equilibrium.

Proof. See Appendix.

5 PPP Rules Defined in Terms of The
CPI-Inflation

In this section we analyze a different type of PPP rule. We
study rules whereby the government in response to an increase in
the CPI-inflation, increases the nominal depreciation rate. The
motivation to consider this type of rule is twofold. First, from an
empirical point of view, Calvo et al. (1995) mention that starting
in 1968, Brazil's government implemented a rule by which the
exchange rate was adjusted as a function of the difference between
domestic and U.S. inflation. In addition, between 1985 and 1992,
Chile used an exchange rate band whose trend was determined by the
difference between the domestic inflation rate and a measure of the
average inflation in the rest of the world. Second, from a
theoretical point of view, Dornbusch (1980, 1982) conceives PPP
rules as a means to introduce the necessary real flexibility to
cope with intrinsic (fundamental) uncertainty in a world that faces
nominal rigidities. He defines a PPP rule as a function whereby the
nominal exchange rate is positively linked to the domestic price
index.

We try to capture the aforementioned stylized facts and some of
the flavor of Dornbusch's work by defining a rule whereby the
nominal depreciation rate is positively linked to the difference
between the domestic CPI inflation () and the foreign CPI-inflation.33 However note that since in our
analysis the foreign variables are considered exogenous and
constant, then the specification of the PPP rule reduces to

(53)

where is a continuous
function.

As before we proceed in the following way. First we will prove
that such rule may induce aggregate instability in the economy by
generating multiple equilibria and opening the possibility of
sunspot equilibria. Specifically we will study and disentangle the
conditions under which this rule leads to real indeterminacy or to
real determinacy. Second, we will study the ``learnability''
properties not only of the fundamental solution but also of the
common factor representation of stationary sunspot equilibria.

The following proposition summarizes the conditions under which
the aforementioned PPP rule induces either real determinacy or real
indeterminacy in the model.

Proposition 5Suppose the government follows a PPP rule given by
with
Let
and be defined as in (41) and define
.

a)Ifthen there is real indeterminacy.

b)If
eitherorthen there is real determinacy.

c)Ifthen there exists no equilibrium.

Proof. See Appendix.

Proposition 5 suggests that multiple equilibria are also
possible for PPP rules that depend on the current CPI-inflation. In
particular it points out that a necessary condition for these rules to cause real
indeterminacy is that the response coefficient to the CPI-
inflation be less than one. That means that in response to a one
percent increase in the CPI-inflation rate, the government raises
the nominal devaluation rate in less than one percent.
Interestingly such response seems to be feasible in the practice of
economic policy. However in order to generate real indeterminacy
the nominal depreciation response coefficient,
must be
above a threshold,
which in turn depends on the structural parameters that affect
and Therefore, as before, we proceed
studying analytically and numerically how varying some structural
parameters of the model affects the previously mentioned threshold.
In particular we focus our analysis on the degree of openness of
the economy, , and
the degrees of monopolistic competition, and price stickiness,
, in the non-traded
sector. The results are presented in Corollary 2.

Corollary 2Suppose the government follows a PPP rule given bywith. Letbe defined as in Proposition 5 and
satisfythena),
b)andc).

Proof. See Appendix.

Using Proposition 5 and Corollary 2 we can breakdown the effects
of varying some of the structural parameters of the model and the
PPP rule response coefficient to CPI-inflation
on the
determinacy of equilibrium analysis. In fact, when
we can
conclude the following. Given the PPP rule response coefficient to
CPI-inflation, the less open the economy is (the lower is), the more likely that the
PPP rule will induce aggregate instability in the economy by
generating multiple equilibria. In addition, given the rule
response coefficient to CPI-inflation and keeping the rest
constant, the higher the degree of monopolistic competition in the
non-traded sector (the higher , the more likely that the rule will lead to real
indeterminacy. Finally under ceteris paribus and given the rule
response coefficient to CPI-inflation, the lower the degree of
price stickiness in the non-traded sector (the lower ), the more feasible that the rule
will induce multiple equilibria.

Under the parametrization of Table 1 we construct Figure 2 that
corroborates these results quantitatively. To some extent it also
validates numerically how likely is that the aforementioned PPP
rule may destabilize the economy by generating multiple equilibria.
Moreover although it is not possible to derive an analytical result
to see how varying the implied nominal depreciation target (
) and
the response coefficient to the CPI-inflation (
) affect the
determinacy of equilibrium, it is possible to evaluate this
quantitatively. In short the bottom right panel of Figure 2
illustrates that given the response coefficient of the rule the
lower the nominal depreciation target is, the more likely is that
the PPP rule will induce real indeterminacy.

It is also important to observe that similar qualitative results
to the ones described in Proposition 5 and Corollary 2 can be
obtained if the PPP rule is defined in terms of the non-traded
goods inflation rate. That is
with
Furthermore it is also possible to prove that a backward-looking
rule defined in terms of the past CPI-inflation rate will avoid
multiple equilibria.34

Figure 2: This figure shows how the local determinacy of equilibrium varies with respect to the PPP rule response coefficient to the
CPI-inflation (
) , the share
of traded goods (),
the degree of monopolistic competition in the non-traded sector (), the degree of price stickiness in the non-traded sector () and the implied nominal depreciation rate target (
). "I" stands for real indeterminacy (multiple equilibria), "D" stands for real
determinacy (a unique equilibrium) and "N" for non-existence of equilibrium. "ES" corresponds to E-Stability.

Description of Figure 2

Figure 2 shows the combinations of the PPP rule response
coefficient to the CPI-inflation and other structural parameters of
the model under which there is a unique equilibrium (real
determinacy) whose Minimal State Variable (MSV) representation is
learnable in the E-stability sense. These combinations are denoted
by `` D-ES''. The figure also shows the combinations of these
parameters under which there are multiple equilibria (real
indeterminacy) and sunspot equilibria whose Common Factor (CF)
representation is learnable in the E-stability sense. We denote
these combinations by `` I-ES''. In addition the figure shows the
combinations that imply the non-existence of an equilibrium denoted
by `` N''. Figure 2 has four panels. We proceed to describe each
panel.

The top-left panel shows the combinations for the PPP rule
response coefficient to the CPI-inflation (
) and the
share of traded goods () under which there is `` D-ES'' or `` I-ES''.
The panel measures
on the
vertical axis with
and on the horizontal axis with
For
there
is `` D-ES'' or `` N''. For
there
is a frontier of combinations of
andthat separates
the combinations of `` D-ES'' from `` I-ES''. This frontier is
convex and has a positive slope. It crosses the
-axis at 0.45
approximately.(`` approximately'' since cannot be zero.) To the
north-west of this frontier we have the combinations for
and
under which there
is `` I-ES''. To the south-east of this frontier we have the
combinations for
and
under which there
is `` D-ES''.

The top-right panel shows the combinations for the PPP rule
response coefficient to the CPI-inflation (
) and the
degree of monopolistic competition () under which there is `` D-ES'', `` I-ES'' or ``
N''. This panel measures
on the
vertical axis with
and on the horizontal axis with
For
and
there is ``
D-ES'' or `` N''. For
and
there is ``
D-ES''. For
there
is a frontier of combinations of
and
that separates the
combinations of `` D-ES'' from `` I-ES''. This frontier is convex,
has a negative slope and goes through
To the north-east of this
frontier we have the combinations for
and
under which there is
`` I-ES''. To the south-west of this frontier we have the
combinations for
and
under which there is
`` D-ES''.

The bottom-left panel shows the combinations for the PPP rule
response coefficient to the CPI-inflation (
) and the
degree of price stickiness () under which there is `` D-ES'', `` I-ES'' or ``
N''. This panel measures
on the
vertical axis with
and on the horizontal axis with
For
and
there is ``
D-ES'' or `` N''. For
and
there is ``
D-ES''. For
there
is a frontier of combinations of
and
that separates the
combinations of `` D-ES'' from `` I-ES''. This frontier is concave,
has a positive slope and goes through
and
approximately.(``
approximately'' since cannot be zero.) To the north-west of this
frontier we have the combinations for
and
under which there
is `` I-ES''. To the south-east of this frontier we have the
combinations for
and
under which there
is `` D-ES''.

The bottom-right panel shows the combinations for the PPP rule
response coefficient to the CPI-inflation (
) and the
depreciation target (
) under
which there is `` D-ES'', `` I-ES'' or `` N''. This panel measures
on the
vertical axis with
and
on the
horizontal axis with
For
and
there is `` D-ES'' or `` N''. For
and
there is `` D-ES''. For
there
is a frontier of combinations of
and
that
separates the combinations of `` D-ES'' from `` I-ES''. This
frontier is concave, has a positive slope and goes through
and
approximately.
To the north-west of this frontier we have the combinations for
and
under
which there is `` I-ES''. To the south-east of this frontier we
have the combinations for
and
under
which there is `` D-ES''.

We proceed by pursuing the learnability analysis. As argued
before this analysis is useful to evaluate the attainability of the
possible unique equilibrium and multiple equilibria induced by the
PPP rule. We use equations (37), (38), (39) and the
log-linearized versions of (8) and
to reduce the model to

(54)

where

(55)

and denotes
in general (non-rational) expectations.

As before in order to pursue the learnability analysis we use
the methodology proposed by Evans and Honkapohja (1999, 2001). We
derive some E-stability conditions and check whether a particular
representation of the equilibrium under analysis satisfies or
violates them.

The following proposition summarizes the results.

Proposition 6Suppose the government follows a PPP rule that is
described by
and
Let
, and
be
defined as in (41) and (55)
respectively and
.

a)Under the
real determinacy conditions specified in Proposition 5 there exists
a unique equilibrium characterized by the fundamental
solution

(56)

whereis uniquely defined by the quadratic
equationandis also uniquely defined
byThis solution is learnable in the
E-stability sense.35

b)Assume
thatUnder the real indeterminacy conditions
specified in Proposition 5 there are stationary sunspot equilibria
described by the common factor representation

whereare unique and correspond to the roots of the quadratic
equationis arbitrary,andis a martingale difference sequence.In particular, the common factor
representation (57) and (58) withandis learnable in the E-stability sense.36

Proof. See Appendix.

Proposition 6 states that when the PPP rule under study induces
a unique equilibrium then this equilibrium represented by the
fundamental solution, also known as the MSV solution, is learnable
in the E-stability sense. This result is important since it means
that given that the rule induces a unique equilibrium then agents
will be able to coordinate on that particular equilibrium and
therefore the economy will converge towards it over time. In
addition as was demonstrated in Proposition 2, it is also possible
to prove for PPP rules defined in terms of the CPI-inflation, that
even under real indeterminacy the fundamental solution is still
E-stable. However under real indeterminacy there are other
equilibria such as stationary sunspot equilibria whose feasibility
is worth evaluating in terms of learnabiliy. Accordingly, the
second part of Proposition 6 shows that a common factor
representation of stationary sunspot equilibria is learnable in the
sense of E-stability. This result is interesting for two reasons.
First, as mentioned before, it suggests that sunspot equilibria
induced by PPP rules are more likely to occur. Second, it warns
policy makers about some of the negative consequences of
implementing PPP rules that respond to inflation. Dornbusch (1980,
1982) conceived PPP rules as a means to introduce the necessary
real flexibility to cope with intrinsic (fundamental) uncertainty
in an economy with nominal rigidities. In contrast our result
points out that such PPP rules may open the possibility of
learnable representations of sunspot equilibria aggravating the
effects of extrinsic (non-fundamental) uncertainty in an economy
with nominal rigidities.

6 Conclusions

In this paper we establish and disentangle the conditions under
which PPP rules lead to real (in)determinacy in a small open
economy that faces nominal rigidities. We find that besides the
specification of the rule, structural parameters such as the share
of traded goods (that measures the degree of openness of the
economy) and the degrees of imperfect competition and price
stickiness in the non-traded sector play a crucial role in the
determinacy of equilibrium.

More importantly to evaluate the relevance of the determinacy
results we also pursue a learnability (E-stability) analysis. We
show that for rules that guarantee a unique equilibrium the
fundamental solution that describes this equilibrium is learnable
in the E-stability sense. Similarly we show that for PPP rules that
open the possibility of sunspot equilibria, some common factor
representations of these equilibria are also E-stable. That is,
agents can coordinate their actions and learn some representations
of stationary sunspot equilibria. In this sense these equilibria
are more likely to occur under PPP rules than previously recognized
and therefore these rules are more prone to cause aggregate
instability in the economy.

Dornbusch (1980, 1982) conceived PPP rules as a means of
introducing the real flexibility necessary to cope with intrinsic
(fundamental) uncertainty in an economy with nominal rigidities.
Our results indicate that PPP rules must be chosen with care in
order to avoid the possibility of ``learnable'' sunspot equilibria
and the associated aggravation of the effects of extrinsic
(non-fundamental) uncertainty. In other words, PPP rules should
satisfy two stability requirements: uniqueness and learnability. On
one hand, the rule should avoid sunspot equilibria that are usually
associated with undesirable properties such as a large degree of
volatility. On the other hand, the rule should guarantee that
agents can indeed coordinate their actions on the equilibrium the
policy makers are targeting and that the economy will not in fact
diverge away from this target.

There are some possible extensions of the analysis presented in
this paper. First, one may consider extending the model to have two
traded goods: a domestic one and a foreign one. This will enrich
the analysis making the model more similar to the ones in Dornbusch
(1980, 1982). Under this set-up one can explore how our results may
vary when the government responds to different measures of
inflation in the PPP rule. Second one may study how our determinacy
and learnability of equilibrium results may be affected by
following the approach by Preston (2003). That is, instead of
imposing the assumption of non-rational expectations on the derived
log-linearized model, we may impose this assumption as a primitive
one of the model. This assumption implies that agents do not have a
complete economic model with which to derive true probability laws
since they do not know other agents' tastes and beliefs. In this
case agents solve multi-period decision problems whereby their
actions depend on forecasts of macroeconomic conditions many
periods into the future. We leave these extensions for further
research.

7 Appendix

Lemma 1In
alinearized system of difference equations
whose matrix is denoted byand whose characteristic
equation corresponds to()=if
either a)
, or b)
or c) then the system
displays real eigenvalues.

Proof. First we recall from Azariadis (1993) that a
sufficient condition for such a linearized system to have real
eigenvalues is that
. Then to
prove a) is trivial.

To prove b) we start by noting that
means that
But this
implies that
that with the aforementioned sufficient condition in turn leads to
. Hence the eigenvalues are real.

To prove c) we point out that
means that
But this
implies that
that in turn leads to
. Hence the eigenvalues are real.

7.1 Proof of Proposition 1

Proof. To prove all the parts of the proposition we use
(37), (38), (39) and (40) to derive the
following system

where

is the forecast error for the non-traded goods inflation defined as
and where the forms of and are
omitted since they are not needed in what follows. For the the
proof of the proposition it is convenient to pursue the determinacy
analysis for the system written in the form
where
instead of
. It will reduce tremendously the number of cases that have
to be analyzed.37 For this
linearized system we have that the trace of , the determinant of , and the characteristic polynomial
associated with
correspond to

respectively. Using these expressions we obtain

where

All these expressions together with
and
and the
assumptions about the structural parameters allows us to observe
that
and
By Lemma 1, this last inequality implies that the eigenvalues are
real.

To prove part a) we proceed as follows. Observe that since
then
.
Hence from
and
we can conclude that
This in conjunction with
and
imply
that the system has two explosive eigenvalues
and
(which means
that
and
Thus the steady state is a source (See Azariadis, 1993). Since
is
the only non-predetermined variable of the system then by Blanchard
and Kahn (1980) we conclude that the model displays real
indeterminacy.

To prove b) we note that from
and
we can conclude
that
This in tandem with
and
imply
that the system has one explosive eigenvalue
and one
non-explosive eigenvalue
Hence the
steady state is a saddle path (See Azariadis, 1993). Since
is
the only non-predetermined variable of the system then by Blanchard
and Kahn (1980) we conclude that the model displays real
determinacy.

Finally to prove c) we use the fact that
implies that
and therefore
Then the analysis pursued to prove part b) follows.

7.2 Proof of Corollary 1

Proof. First use the steady state description and the
definition of and
given by (41) to calculate

where we use the assumptions about the structural parameters to
determine the sign of the derivatives. Next we use these results
and the definition of
given
in Proposition 1, together with the assumptions about the
structural parameters to derive

where

7.3 Proof of Proposition 2

Proof. To prove a) we proceed in the following way.
First, we rewrite the first equation in (45) as

(59)

where is the lag operator,
and
is a martingale difference sequence. The associated characteristic
equation of (59) is

(60)

whose roots are denoted by
with
. Second we
establish a relationship between the roots
and the
roots
in the
proof of Proposition 1. In particular it is straightforward to show
that
and that

(61)

where was defined in the
proof of Proposition 1.

Third, using these relationships and the proof in Proposition 1,
it is trivial to show that under real determinacy, the unique
equilibrium of the model (45)
characterized by the fundamental solution (47)
with
is in
fact a solution of the model (45).

Fourth, we derive the E-stability conditions. Consider the model
(45) and assume that the agents follow a
perceived law of motion (PLM) that in this case of real determinacy
corresponds to the fundamental solution

Iterating forward this law of motion and taking expectations we
obtain
; using this to eliminate all the forecasts in the model
(45) and assuming that agents know
when they
make their forecasts, we can derive the implied actual law of
motion (ALM)

Then we obtain the T-mapping
whose fixed points correspond to the rational expectations
equilibrium with
satisfying
satisfying (60) and satisfying
. Note that since
then
. Moreover since the matrix of derivatives
is block triangular then
it is simple to show that its eigenvalues are
and
which in turn mean that the E-stability conditions
correspond to

Finally recall that in the proof of Proposition 1 we derived
that real determinacy is associated with one explosive eigenvalue
and one
non-explosive eigenvalue
Then using
this and
we can infer that
and either
or
Moreover using (61) we can rewrite the conditions (62) as

Then defining
Proposition 2 follows since it is simple to see that the
E-stability conditions are satisfied given
,
and either
or

7.4 Proof of Proposition 3

Proof. This proof builds on Evans and McGough (2003).
First, we rewrite the model (45) as

(65)

where is the lag operator,
and
is a martingale difference sequence. The associated characteristic
equation of (65) is

(66)

whose roots are denoted by
with
. Note that
.
Recall that in the proof of Proposition 1 we argued that real
indeterminacy was associated with the two explosive eigenvalues
and
Then using
this and
we can infer that
and
Moreover using (61) and the definitions
and
in
(46) we obtain
Using the conditions of real indeterminacy,
and
it is simple to derive
that
Then using Lemma 1 we can conclude that the roots are real.
In addition without loss of generality we can assume that the roots
are
and

Second, we point out that following Propositions 3 and 4 in
Evans and McGough (2003), it is simple to prove that the process
is a
rational expectations equilibrium in (45)
with
, if and
only if there is a martingale difference sequence
such that
solves
(48) with (49).

Third, we derive the E-stability conditions adapting the
analysis of Evans and McGough (2003). In particular, note that we
assume that agents knows
when
making the forecast
Consider the stochastic model (45) and
suppose that the agents follow a perceived law of motion (PLM) such
as

where
is a martingale difference sequence. Iterating forward these laws
of motion and taking expectations we obtain
; using this to eliminate all the forecasts in the model
(45) and assuming that agents know
when they
make their forecasts, we can derive the implied actual law of
motion (ALM)

Then we obtain the T-mapping

whose fixed points correspond to the rational expectations
equilibrium with and satisfying (66),
respectively and
38 Since the matrix of derivatives
is
block triangular then it is simple to show that the eigenvalues
correspond to
and
which in
turn means that the E-stability conditions reduce to

Moreover using (66) and noting that
and
we can
rewrite these conditions as

(67)

However note that for the last E-stability condition, it is
always true that for the common factor representation we have that
either
and therefore
implying
or,
and therefore
implying
Hence the differential equation for
is
Using a similar argument to the one developed in Evans and
Honkapohja (1992) it is possible to show that as either
or
then
converges to a finite value. This means that the only stability
conditions that are required to be checked are
and

Fourth, we recall our result from the beginning of this proof
that states that under real indeterminacy the roots are
and

To prove part a) we use the fact that for
we
have that the E-stability conditions
and
become
and
Since
and
it is clear that these E-stability conditions are satisfied. Hence
the common factor representation (48) and
(49) with
and
is learnable in the
E-stability sense.

To prove part b) we utilize the fact that for
we
have that the E-stability condition
is
clearly not satisfied for
given that
implies

7.5 Proof of Proposition 4

Proof. To prove a) first we write the characteristic
equation associated with (50) as

where
and
are
defined in (51). Second using this, and
and
and the
assumptions about the structural parameters we can derive that

where
. Observe that since
by
Lemma 1 we know that the eigenvalues are real.

Third note that if either
and
or
and
then
Using this and the previous results that
and that the eigenvalues are real we may infer that the steady
state is a saddle with one of the eigenvalues falling in
() and the other
one falling in () as explained in Azariadis (1993). Given that
is the
only predetermined variable then by Blanchard and Kahn (1980) we
conclude that the model displays real determinacy.

Fourth it is straightforward to prove that the fundamental
solution
is in fact a solution of (50).
Fifth we prove that this fundamental solution is E-stable. In order
to do so we need to derive the E-stability conditions. The
procedure to derive them is exactly the same as the procedure
followed in the proof of Proposition 2. In fact the conditions are
the same as the ones previously derived. Here we present only the
conditions. The E-stability conditions are

It is simple to see that the E-stability conditions are
satisfied given
,
and
Then the fundamental
solution is learnable in the E-stability sense and statement a) of
this proposition follows.

To prove b) it is enough to note that if
and
then we can derive
that
.
Using this and the previous results that
and that the eigenvalues are real we may infer from Azariadis
(1993) that the steady state is a source with one eigenvalue
falling in (
) and the
other in ().
Since
is the
only predetermined variable then by Blanchard and Kahn (1980) we
conclude that the model displays no equilibrium for this case.

7.6 Proof of Proposition 5

Proof. To prove all the parts of the proposition we use
(37), (38), (39) and the log-linearized versions of (8) and
to derive the following
system

where

is the forecast error for the non-traded goods inflation defined as
and where the forms of and are
omitted since they are not needed in what follows. For the the
proof of the proposition it is convenient to pursue the determinacy
analysis for the system written in the form
where
instead of
. It will reduce tremendously the number of cases that have
to be analyzed.39 For this
linearized system we have that the trace of , the determinant of , and the characteristic polynomial
associated with
correspond to

respectively. Using these expressions we obtain

where
In this proof we will use the facts that and as well as the constraints
imposed on the structural parameters listed in the description of
the model.

To prove part a) we proceed as follows. Observe that since
and
then
and therefore
By Lemma 1, this last inequality implies that the eigenvalues are
real. In addition since by assumption
(and
), then we can infer
that
It is straightforward to show that
and
imply that
a result we will use
later Hence
in tandem with
help us to conclude that the system has two explosive eigenvalues
and
and that the
steady state is a source as explained in Azariadis (1993). This
means that
and
Since
is
the only non-predetermined variable of the system then by Blanchard
and Kahn (1980) we conclude that the model displays real
indeterminacy.

To prove b) consider the assumption
We have to take into account two cases:
and
For
we have that since by
assumption
then
By Lemma 1, this last inequality implies that the eigenvalues are
real. In addition since by assumption
then we can infer
that
This in tandem with
imply that the steady state is a saddle point with one explosive
eigenvalue
and one non-explosive
eigenvalue
as shown in Azariadis (1993).

For
we have that since by
assumption
then
In addition since by assumption
then we can infer
that
which in turn implies, by Lemma 1, that the eigenvalues are real.
Since
and
then the steady state is a saddle point and the system has one
explosive eigenvalue
and one non-explosive
eigenvalue
as explained in Azariadis (1993).

Now consider the assumption
Using this we can infer that in any case we have that since
then
But this implies
that
By Lemma 1, we derive that the eigenvalues are real. In addition
since by assumption
then we can infer
that
This in tandem with
imply that the steady state is a saddle point with one explosive
eigenvalue
and one non-explosive
eigenvalue
as shown in Azariadis (1993)

Therefore under either
or
the steady state is a saddle path. Since
is
the only non-predetermined variable of the system then by Blanchard
and Kahn (1980) we conclude that the model displays real
determinacy.

Finally to prove c) we start by observing that
Hence
.
These inequalities imply that
and
Then we have to consider two cases:
and
For the
first case if
then since
we have that
By Lemma 1 this implies that
the roots are real. Moreover utilizing this and
and
we can infer that the steady state is a sink with two non explosive
eigenvalues
and
as explained in Azariadis(1993)

On the other hand, for the second case we have that since
and
then
(provided that
Using
this and
and
we can conclude from Azariadis (1993) that regardless of whether
the eigenvalues are real or complex the steady state is a source
with two non explosive eigenvalues
and

Therefore in both cases we have concluded that the steady-state
is a source. Hence
and
Since
is
the only non-predetermined variable of the system then by Blanchard
and Kahn (1980) we conclude that the model displays no equilibrium.

7.7 Proof of Corollary 2

Proof. First, observe that
implies that
Second, from the Proof of
Corollary 1 we have that
and
Use all these
inequalities in tandem with

where
and
to derive a), b) and c).

7.8 Proof of Proposition 6

Proof. First, observe that the characteristic equation
associated with (54) is

(68)

whose roots are denoted by
with
. Second, we
establish a relationship between these roots
and the
roots
in the
proof of Proposition 5. In particular it is straightforward to show
that
and that

where was defined in
Proposition 5.

Third we use these relationships to prove a). Since under real
determinacy in the proof of Proposition 5 we have that either
and
or
and
then we can conclude that
and either
or

Fourth, it is simple to show that under real determinacy, the
unique equilibrium of the model (54)
characterized by the fundamental solution (56)
with
is in
fact a solution of the model (54).

Fifth, we derive the E-stability conditions. However the
procedure is the same as the one explained in the proof of
Proposition 2. We only rename
and
as
and
respectively. Then following the proof of Proposition 2, it is
straightforward to prove that the unique equilibrium of the model
(54) characterized by the fundamental
solution (56) with
and
either
or
is E-stable.

Sixth to prove b) we start by noting that in (68) we have that
. Next we recall that in the proof of a) in Proposition 5 we
argued that real indeterminacy is associated with two non-explosive
eigenvalues
and
Then using
this and
help us to infer
that
and
Furthermore under real indeterminacy it is easy to show that
. Hence
which in turn means that the roots are real Then without loss of generality we
can assume that the roots are
and

Seventh, we point out that following Propositions 3 and 4 in
Evans and McGough (2003), it is simple to prove that the process
is a
rational expectations equilibrium of (54)
with
, if and
only if there is a martingale difference sequence
such that
solves
(57) with (58).
Henceforth we focus on proving the learnability of the common
factor representation. However the proof is similar to the proof
for Proposition 3. We just have to rename
and
as
and
respectively, and follow the steps. Then statement b) follows.

Footnotes

2. This idea is not specific to PPP
rules. In fact the idea that a rule that leads to indeterminacy of
equilibrium may be seen as undesiderable has been emphasized by
recent studies about interest rate rules. See Benhabib,
Schmitt-Grohé and Uribe (2001), Carlstrom and Fuerst (2001),
Clarida, Gali and Gertler (2000), Rotemberg and Woodford (1999) and
Woodford (2003) among others. Return to text

3. Bullard and Mitra (2002) have
emphasized the importance of this point in the interest rate rule
literature. Return to text

4. Henceforth we will use the terms
`` learnability'', `` E-stability'' and `` expectational
stability'' interchangeably in this paper. Return to text

5. Evans and Honkapoja (1999, 2001)
have argued that a unique equilibrium and sunspot equilibria are
not `` fragile'' if they are learnable in the sense of E-stability.
Technically what they propose is to assume that agents in the model
initially do not have rational expectations but are endowed with a
mechanism to form forecasts using recursive learning algorithms and
previous data from the economy. Then they develop some E-stability
conditions which govern whether or not a given rational
expectations equilibrium is aymptotically stable under least
squares learning. Return to
text

6. From now on we will use the terms
`` multiple equilibria'' and `` real indeterminacy'' (a `` unique
equilibrium'' and `` real determinacy'') interchangeably. By real
indeterminacy we mean a situation in which the behavior of one or
more (real) variables of the model is not pinned down by the model.
This situation implies that there are multiple equilibria and opens
the possibility of the existence of sunspot equilibria. Return to text

7. This fundamental solution is also
well-known as the Minimal State Variable solution. See McCallum
(1983). Return to text

8. The common factor representation
is an alternative representation of a Rational Expectations
Equilibria. See Evans and Honkapoja (1986). Return to text

9. Dornbusch (1980,1982) uses a
Mundell and Fleming small open economy model with sticky wages
á la Taylor and finds that the
aformentioned PPP rule affects the output price-level stability
trade-off through two different channels. On one hand, it tries to
maintain constant the real exchange rate stabilizing net exports
and therefore the demand side. On the other hand, it affects the
supply side by its effect on the price of imported intermediate
goods. Dornbusch shows that in such a model if the economy is hit
by supply shocks then the price volatility always increases with
tighter PPP rules. If the demand channel dominates the supply
channel then the PPP rule reduces the volatility of output. But if
the supply channel dominates the demand channel then the volatility
of output is increased. Return to
text

10. The set-up of this model is very
similar to Uribe (2003) and Zanna (2003a). However we endogenize
labor in both sectors and introduce technology shocks. Moreover we
use specific functional forms to be able to convey the main message
of this paper. In particular we assume separability in terms of
both types of consumption. A CES utility function will not affect
the qualitative results of this paper but will make the derivation
of our analytical results cumbersome. Return to text

11. For the first part of the paper
we will assume that agents have rational expectations. However for
the E-stability analysis we will relax this assumption. Return to text

13. We follow Woodford (2003) to
construct the budget constraint of the representative
agent. Return to text

14. This price is equal to its
output cost (=1) plus a term that is a function of the opportunity
cost of holding wealth in monetary form. Return to text

15. Note that as a consequence of
the aforementioned contingent claims that completely span the
uncertainty about productivity shocks the model abstracts from
wealth effects due to current account imbalances. In this respect
the model is similar to the ones in Clarida, Gali and Gertler
(2001) and Gali and Monacelli (2004). Return to text

16. We would have derived a similar
augmented Phillips curve if we had follow Calvo's (1983)
approach. Return to text

17. Note that the spirit of the PPP
rule and the assumption that is sticky imply that is a predetermined variable. As
a consequence assuming that and are given corresponds to assume that
is
given which in turn avoids the possibility of nominal
indeterminacy. Return to
text

18. Observe that we have not
included equation
.
The reason is that
does
not affect the other equations. It may affect the current account.
But as in Clarida et al. (2001) and Gali and Monacelli (2004) we
are abtracting from wealth effects due to current account
imbalances. Return to text

20. It is important to remember that
in the log-linearized set-up all the variables are expressed as
deviations from their steady state level. Return to text

21. It is possible to do the same
exercise with respect to other structural parameters such as the
share of labor in the production function (
). These
results are available upon request. We focus our analyzis on the
share of traded goods () because this is a particular feature of open
economies. We also concentrate the analysis on the degrees of
monopolistic competition () and price stickiness () in the non-traded sector because
these parameters capture an important asymmetry between the traded
and non-traded sectors. Return to
text

22. Note that for this exercise we
do not need to assign values to all the parameters. We only present
the parametrization of the relevant parameters. Return to text

23. There is no clear consensus
about the value that this parameter must take in emerging
economies. One of the reasons is the lack of studies that have
tried to estimate Phillips curves for these economies and that may
give information about possible values for this parameter. Even for
an industrialized economy such as Canada, this parameter varies
between 2.80 and 44.07, depending on the model specification (type
of nominal and real rigidities). See Dib (2001). Return to text

25. The idea of expectation driven
fluctuations dates back to Keynes (1936). Return to text

26. It is important to observe that
for models with multiple stationary equilibria this statement lacks
of technical formality. As pointed out by Evans and McGough (2003)
for a model with multiple equilibria, a rational expectations
equilibrium may have different representations. Therefore one
should not strictly speak of learnable rational expectations
equilibrium, but whether a rational expectations equilibrium
representation is learnable (E-stable). Return to text

27. The Minimal State Variable (MSV)
solution according to McCallum (1983). Return to text

28. Observe that this definition
suggests that to prove E-stability of a fixed point corresponds to
prove that all the eigenvalues of the matrix of derivatives
are less than 1. Return to text

29. The proof of learnability of
common factor representations of sunspot equilibria in this paper
also goes over this methodology. Return to text

30. Note that strictly speaking
since we are assuming that the economy starts at and we have an initial condition
for
then in
the common factor representation of a rational expectations
equilibrium
equation
(48) should be written as

for
and
arbitrary. However
since
then as
the
solution
converges
to a process that satisfies (48). Return to text

31. Note that
NOT introducing the constant in the perceived law of motion, such that
is not innocuous for statement b) of the proposition. In
this case the common factor representation with
and
becomes
E-stable. Return to
text

32. Forward-looking PPP rules
defined in terms of the expected future real exchange rate still
open the possibility of sunspot equilibria as shown in Zanna
(2003b). Return to text

35. Under real indeterminacy it is
also possible to prove that there is an equilibrium characterized
by fundamental solution (56) with
which
is E-stable Return to text

36. Excluding a constant
in the
perceived law of motion such that
is not innocuous for this part of the proposition. In this
case the common factor representation with
and
is also
E-stable. Return to text

37. One can analyze either of the
forms because the eigenvalues of correspond to
where
are the
eigenvalues of Return to
text